Toeplitz operators with locally integrable symbols on Bergman spaces of bounded simply connected domains

Abstract

We study the boundedness and compactness of generalized Toeplitz operators with locally integrable symbols on Bergman spaces where is a bounded simply connected domain with smooth boundary. We give sufficient conditions for boundedness and compactness of in terms of “averages” of symbol a over certain Cartesian squares. The main tool in the proof is the Whitney covering: is decomposed into union of countably many squares whose side lengths are comparable to the boundary distance. If a is nonnegative, we show that the given conditions are also necessary.